Normal distribution

• SAT scores follow a nearly normal distribution with a mean of 1500 points and a standard deviation of 300 points.

• ACT scores also follow a nearly normal distribution with mean of 21 points and a standard deviation of 5 points.

• Nel scored 1800 points on their SAT

• Sian scored 24 points on their ACT.

• Who performed better?

Setup

import pandas as pd
import numpy as np
import scipy.stats as st

import altair as alt

alt.data_transformers.disable_max_rows()

DataTransformerRegistry.enable('default')

Data

Generate data

sat_mean = 1500
sat_sd   = 300
act_mean = 21
act_sd   = 5

nel_sat = 1800
sian_act = 24 

np.random.seed(0)

sat = np.random.normal(sat_mean, sat_sd, 100000)
act = np.random.normal(act_mean, act_sd, 100000)

df = pd.DataFrame({"sat": sat, "act":act})

Data overview

df.head()

	sat	act
0	2029.215704	18.581013
1	1620.047163	27.440285
2	1793.621395	20.350606
3	2172.267960	20.009608
4	2060.267397	19.327562

df.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 100000 entries, 0 to 99999
Data columns (total 2 columns):
 #   Column  Non-Null Count   Dtype  
---  ------  --------------   -----  
 0   sat     100000 non-null  float64
 1   act     100000 non-null  float64
dtypes: float64(2)
memory usage: 1.5 MB

Data visualization

chart = alt.Chart(df).transform_density(
    'sat',
    as_=['sat', 'density'],
).mark_area().encode(
    x="sat:Q",
    y='density:Q',
)

nel = alt.Chart(pd.DataFrame({
  'value': nel_sat,
  'color': ['orange']
})
).mark_rule(
  strokeDash=[5, 5], 
  strokeWidth=3
).encode(
  x='value:Q',
  color=alt.Color('color:N', scale=None)
)

chart + nel

chart = alt.Chart(df).transform_density(
    'act',
    as_=['act', 'density'],
).mark_area().encode(
    x="act:Q",
    y='density:Q',
)

sian = alt.Chart(pd.DataFrame({
  'value': sian_act,
  'color': ['orange']
})
).mark_rule(
  strokeDash=[5, 5], 
  strokeWidth=3
).encode(
  x='value:Q',
  color=alt.Color('color:N', scale=None)
)

chart + sian

Standardizing with Z scores

z_nel = (nel_sat - sat_mean) / sat_sd
z_nel

1.0

z_sian = (sian_act - act_mean) / act_sd
z_sian

0.6

Normal probability calculations

Nel’s percentile is the percentage of people who earned a lower SAT score than Nel.

nel_percentile = st.norm.cdf(z_nel)
print(f"Nel’s percentile: {nel_percentile:.2}")

Nel’s percentile: 0.84

Determine the proportion of SAT test takers who scored better than Nel on the SAT.

sian_percentile = st.norm.cdf(z_sian)
round(sian_percentile, 2)

0.73

We can also find the Z score associated with a percentile. For example, to identify Z for the 80th percentile, we use st.norm.ppf() which identifies the quantile for a given percentage.

st.norm.ppf(.80)

0.8416212335729143